Proposition 3.2.6 (Homotopy invariance of K0)

Let and be unital -algebras

  1. If are homotopic -homomorphisms, then .
  2. If and are homotopy equivalent, then is isomorphic to More specifically, if is a homotopy, then and are isomorphisms, and .

Proof:

  1. Let be a point-wise continuous path of -homomorphisms connecting and . Extend this path to a pointwise continuous -homomorphisms for each .
    For every projection the path is continuous (pointwise continuous) and so . This shows that The map sends projections to projections, and these two projections are homotopic, so by shifting up to matrices, we get that they are K0 equivalent???
  2. Follows from Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras) parts 1 and 2